Sachin lent out Rs 60000 in two parts, first at 4% and the second at 10% interest. The yeariy average interest comes out to be 6.4%. What are the amounts (in Rs) that were lent at 4% and 10% respectively?

Options

A. 40000, 20000
B. 48000, 12000
C. 36000, 24000
D. 32000, 28000

Answer

To solve the problem, we need to determine how much Sachin lent at 4% and how much he lent at 10%, given that the total amount lent is Rs 60,000 and the average interest rate is 6.4%.

### Step-by-Step Solution:

1. **Define Variables**:
Let:
- \( x \) = amount lent at 4%
- \( y \) = amount lent at 10%

According to the problem:
\[ x + y = 60000 \quad \text{(1)} \]

2. **Set Up the Average Interest Equation**:
The average interest can be calculated using the formula:
\[ \text{Average Interest} = \frac{\text{Total Interest}}{\text{Total Principal}} \]
The total interest earned from both parts can be expressed as:
\[ \text{Interest from } x = \frac{4}{100} \cdot x \]
\[ \text{Interest from } y = \frac{10}{100} \cdot y \]
Therefore, the total interest is:
\[ \text{Total Interest} = \frac{4}{100}x + \frac{10}{100}y \]
The average interest rate is given as 6.4%, so we can set up the equation:
\[ \frac{\frac{4}{100}x + \frac{10}{100}y}{60000} = \frac{6.4}{100} \quad \text{(2)} \]

3. **Multiply Through by 60000**:
To eliminate the denominator, multiply both sides of equation (2) by 60000:
\[ \frac{4}{100}x + \frac{10}{100}y = \frac{6.4}{100} \cdot 60000 \]
Simplifying the right side:
\[ \frac{6.4 \cdot 60000}{100} = 3840 \]
Thus, we have:
\[ \frac{4}{100}x + \frac{10}{100}y = 3840 \]
Multiplying through by 100 to eliminate the fractions:
\[ 4x + 10y = 384000 \quad \text{(3)} \]

4. **Solve the System of Equations**:
Now we have two equations:
- From (1): \( x + y = 60000 \)
- From (3): \( 4x + 10y = 384000 \)

We can express \( y \) in terms of \( x \) from equation (1):
\[ y = 60000 - x \]
Substituting this into equation (3):
\[ 4x + 10(60000 - x) = 384000 \]
Simplifying:
\[ 4x + 600000 - 10x = 384000 \]
Combining like terms:
\[ -6x + 600000 = 384000 \]
Rearranging gives:
\[ -6x = 384000 - 600000 \]
\[ -6x = -216000 \]
Dividing by -6:
\[ x = 36000 \]

5. **Find \( y \)**:
Now substitute \( x \) back to find \( y \):
\[ y = 60000 - 36000 = 24000 \]

### Final Answer:
- Amount lent at 4%: Rs 24000
- Amount lent at 10%: Rs 36000